
\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}


\textwidth 4.5in
\hoffset 1in
\voffset 2in
\textheight = 8in

\oddsidemargin = 0.0 in
\evensidemargin = 0.0 in
\topmargin = 0.0 in
\headheight = -2.0 in
\headsep = -1.0 in
\parskip = 0.2in
\parindent = 0.0in


\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}

\def\today{}



\title{About the Catenoid Fence}
\author{Hermann Karcher}
\begin{document}
\maketitle




    These singly periodic surfaces are parametrized (aa) by
rectangular tori; our lines extend polar coordinates around
the two punctures to the whole torus. The surfaces look
like a fence of catenoids, joined by handles; they were made by
Karcher and Hoffman, responding to the suggestive skew 4-noids.
The morphing parameter aa is the modulus (a function of the
length ratio) of the rectangular torus. Formulas are from [K2]


[K2]  H. Karcher, Construction of minimal surfaces, in ``Surveys in 
      Geometry'', Univ. of Tokyo, 1989, and Lecture Notes No. 12, 
      SFB 256, Bonn, 1989, pp. 1--96.


  For a discussion of techniques for creating minimal surfaces with 
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].

[KWH]  H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and
         the minimal surfaces that led to its discovery, in ``Global Analysis
         in Modern Mathematics, A Symposium in Honor of Richard Palais' 
         Sixtieth Birthday'', K. Uhlenbeck Editor, Publish or Perish Press, 1993

[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab, 
           Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991



 \end{document}